Rissanen's Stochastic Complexity in Financial Markets

Shu-Heng Chen and Ching-Wei Tan
Department of Economics, National Chengchi University, Taiwan
ChChen@cc.nccu.edu.tw
g4258506@grad.cc.nccu.edu.tw

Abstract

Recent econometric studies on the dynamics of stock market prices contribute to the discovery of the following two phenomena. First, despite the possibility that there exist patterns or regularities in the history of stock prices, they are highly complex or nonlinear and cannot be exploited by standard statistical techniques. Second, those patterns tend to be short-lived. The first phenomenon is established by the failures to reject the nonlinearity of rates of return of stock indices often seen in the nonlinearity tests such as the BDS test. The second phenomenon is highlighted by the nonexistence of the time-invariant structure of the rates of return or the long-memory processes of rates of return, such as the range-scale (R-S) analysis. There is little doubt that these two properties are crucial for the validity of the efficient market hypothesis (EMH). Because if these properties are true, then the chance that investors can devise an investment strategy to yield abnormal profits on the basis of an analysis of past price patterns is low, and so the validity of the EMH can sustain.

Given the importance of these two properties, the purpose of this paper is to construct a dynamic measure for the observability of patterns and the duration of patterns, based on the concept of stochastic complexity developed by Rissanen. By ``dynamic measure", we mean that the measure itself is a function of time. This enables us to see the whole process in which patterns with different degrees of observability actually appeare, survive for a while, and die. This will not only tell us whether the market is efficient in general but will also tell us more specifically about its efficiency in different periods of time.

To do this, Rissanen's MDLP (minimum description length principle) is introduced. The MDLP is an approximation for Kolmogorov complexity which measures the complexity of a set of data by the length of the shortest Turing machine program that will generate the data. The measure is well-defined, but not practically computable. The MDL developed by Rissanen is a way to approximate this uncomputable measure by replacing the universal Turing machine with a class of probabilistic models. We then construct the measure by transforming the orginal sequence of rate of return Rt into a 0-and-1 sequence based on the sign of Rt. Then MDL is computed for each of the 50 consecutive observations in the 0-and-1 sequence by choosing the Bernoulli class and Markov class as our model classes. The data used in this paper concern the daily rate of return of the Taiwan Stock Price Index (TAIEX) and S&P 500 Index, which are available from the EPS database. From 1/5/71 to 1/27/94, there are 6'677 observations in the Taiwan dataset and 5831 in the S&P 500 dataset. Our measure shows that, during this period, while 91% of the time the U.S. stock market is efficient, only 73% of the time can this be said of the Taiwan stock market. This study also indicates that the regularities existing in the history of Taiwan stock prices are much stronger and can survive longer than those existing in the U.S. stock market. Therefore, by the complexity measure, U.S. stock prices do exhibit more complex behaviour than their Taiwan counterparts.

By Monte Carlo simulation, we also find that the U.S. stock market is still rather efficient in comparison with random walks. The Taiwan stock market, however, is not.

Keywords: Kolmogorov's complexity, Stochastic complexity, Minimum description length principle, Efficient market hypothesis, Monte Carlo experimentation.